Lecture Notes for Chapter 2 Introduction to Data Mining Data Mining: Data Lecture Notes for Chapter 2 Introduction to Data Mining by Tan, Steinbach, Kumar

What is Data? Collection of data objects and their attributes An attribute is a property or characteristic of an object Examples: eye color of a person, temperature, etc. Attribute is also known as variable, field, characteristic, or feature A collection of attributes describe an object Object is also known as record, point, case, sample, entity, or instance Attributes Objects

Attribute Values Attribute values are numbers or symbols assigned to an attribute Distinction between attributes and attribute values Same attribute can be mapped to different attribute values Example: height can be measured in feet or meters Different attributes can be mapped to the same set of values Example: Attribute values for ID and age are integers But properties of attribute values can be different ID has no limit but age has a maximum and minimum value

Types of Attributes There are different types of attributes Nominal Examples: ID numbers, eye color, zip codes Ordinal Examples: rankings (e.g., taste of potato chips on a scale from 1-10), grades, height in {tall, medium, short} Interval Examples: calendar dates, temperatures in Celsius or Fahrenheit. Ratio Examples: temperature in Kelvin, length, time, counts

Properties of Attribute Values The type of an attribute depends on which of the following properties it possesses: Distinctness: =  Order: < > Addition: + - Multiplication: * / Nominal attribute: distinctness Ordinal attribute: distinctness & order Interval attribute: distinctness, order & addition Ratio attribute: all 4 properties

Discrete and Continuous Attributes Discrete Attribute Has only a finite or countably infinite set of values Examples: zip codes, counts, or the set of words in a collection of documents Often represented as integer variables. Note: binary attributes are a special case of discrete attributes Continuous Attribute Has real numbers as attribute values Examples: temperature, height, or weight. Practically, real values can only be measured and represented using a finite number of digits. Continuous attributes are typically represented as floating-point variables.

Types of data sets Record Graph Ordered Data Matrix Document Data Transaction Data Graph World Wide Web Molecular Structures Ordered Spatial Data Temporal Data Sequential Data Genetic Sequence Data

Important Characteristics of Structured Data Dimensionality Curse of Dimensionality Sparsity Only presence counts Resolution Patterns depend on the scale

Record Data Data that consists of a collection of records, each of which consists of a fixed set of attributes

Data Matrix If data objects have the same fixed set of numeric attributes, then the data objects can be thought of as points in a multi-dimensional space, where each dimension represents a distinct attribute Such data set can be represented by an m by n matrix, where there are m rows, one for each object, and n columns, one for each attribute

Document Data Each document becomes a `term' vector, each term is a component (attribute) of the vector, the value of each component is the number of times the corresponding term occurs in the document.

Transaction Data A special type of record data, where each record (transaction) involves a set of items. For example, consider a grocery store. The set of products purchased by a customer during one shopping trip constitute a transaction, while the individual products that were purchased are the items.

Graph Data Examples: Generic graph and HTML Links

Ordered Data Sequences of transactions Items/Events An element of the sequence

Ordered Data Genomic sequence data

Ordered Data Spatio-Temporal Data Average Monthly Temperature of land and ocean

Data Quality What kinds of data quality problems? How can we detect problems with the data? What can we do about these problems? Examples of data quality problems: Noise and outliers missing values duplicate data

Noise Noise refers to modification of original values Examples: distortion of a person’s voice when talking on a poor phone and “snow” on television screen Two Sine Waves Two Sine Waves + Noise

Outliers Outliers are data objects with characteristics that are considerably different than most of the other data objects in the data set

Missing Values Reasons for missing values Handling missing values Information is not collected (e.g., people decline to give their age and weight) Attributes may not be applicable to all cases (e.g., annual income is not applicable to children) Handling missing values Eliminate Data Objects Estimate Missing Values Ignore the Missing Value During Analysis Replace with all possible values (weighted by their probabilities)

Duplicate Data Data set may include data objects that are duplicates, or almost duplicates of one another Major issue when merging data from heterogeous sources Examples: Same person with multiple email addresses Data cleaning Process of dealing with duplicate data issues

Data Preprocessing Aggregation Sampling Dimensionality Reduction Feature subset selection Feature creation Discretization and Binarization Attribute Transformation

Aggregation Combining two or more attributes (or objects) into a single attribute (or object) Purpose Data reduction Reduce the number of attributes or objects Change of scale Cities aggregated into regions, states, countries, etc More “stable” data Aggregated data tends to have less variability

Sampling Sampling is the main technique employed for data selection. It is often used for both the preliminary investigation of the data and the final data analysis. Statisticians sample because obtaining the entire set of data of interest is too expensive or time consuming. Sampling is used in data mining because processing the entire set of data of interest is too expensive or time consuming.

Sampling … The key principle for effective sampling is the following: using a sample will work almost as well as using the entire data sets, if the sample is representative A sample is representative if it has approximately the same property (of interest) as the original set of data

Types of Sampling Simple Random Sampling Sampling without replacement There is an equal probability of selecting any particular item Sampling without replacement As each item is selected, it is removed from the population Sampling with replacement Objects are not removed from the population as they are selected for the sample. In sampling with replacement, the same object can be picked up more than once Stratified sampling Split the data into several partitions; then draw random samples from each partition

Sample Size 8000 points 2000 Points 500 Points

Sample Size What sample size is necessary to get at least one object from each of 10 groups.

Dimensionality Reduction Purpose: Avoid curse of dimensionality Reduce amount of time and memory required by data mining algorithms Allow data to be more easily visualized May help to eliminate irrelevant features or reduce noise Techniques Principle Component Analysis Singular Value Decomposition Others: supervised and non-linear techniques

Dimensionality Reduction: PCA Goal is to find a projection that captures the largest amount of variation in data x2 e x1

Dimensionality Reduction: PCA Find the eigenvectors of the covariance matrix The eigenvectors define the new space x2 e x1

Dimensionality Reduction: PCA

Feature Subset Selection Another way to reduce dimensionality of data Redundant features duplicate much or all of the information contained in one or more other attributes Example: purchase price of a product and the amount of sales tax paid Irrelevant features contain no information that is useful for the data mining task at hand Example: students' ID is often irrelevant to the task of predicting students' GPA

Mapping Data to a New Space Fourier transform Wavelet transform Two Sine Waves Two Sine Waves + Noise Frequency

Discretization Using Class Labels Entropy based approach 3 categories for both x and y 5 categories for both x and y

Discretization Without Using Class Labels Data Equal interval width Equal frequency K-means

Attribute Transformation A function that maps the entire set of values of a given attribute to a new set of replacement values such that each old value can be identified with one of the new values Simple functions: xk, log(x), ex, |x| Standardization and Normalization

Attribute Transformation Min-max normalization: to [new_minA, new_maxA] Ex. Let income range $12,000 to $98,000 normalized to [0.0, 1.0]. Then $73,600 is mapped to

Similarity and Dissimilarity Numerical measure of how alike two data objects are. Is higher when objects are more alike. Often falls in the range [0,1] Dissimilarity Numerical measure of how different are two data objects Lower when objects are more alike Minimum dissimilarity is often 0 Upper limit varies Proximity refers to a similarity or dissimilarity

Similarity/Dissimilarity for Simple Attributes p and q are the attribute values for two data objects.

Euclidean Distance Euclidean Distance Where n is the number of dimensions (attributes) and pk and qk are, respectively, the kth attributes (components) or data objects p and q. Standardization is necessary, if scales differ.

Euclidean Distance Distance Matrix

Minkowski Distance Minkowski Distance is a generalization of Euclidean Distance Where r is a parameter, n is the number of dimensions (attributes) and pk and qk are, respectively, the kth attributes (components) or data objects p and q.

Minkowski Distance: Examples r = 1. City block (Manhattan, taxicab, L1 norm) distance. A common example of this is the Hamming distance, which is just the number of bits that are different between two binary vectors r = 2. Euclidean distance r  . “supremum” (Lmax norm, L norm) distance. This is the maximum difference between any component of the vectors Do not confuse r with n, i.e., all these distances are defined for all numbers of dimensions.

Minkowski Distance Distance Matrix

Mahalanobis Distance  is the covariance matrix of the input data X For red points, the Euclidean distance is 14.7, Mahalanobis distance is 6.

Mahalanobis Distance Covariance Matrix: C A: (0.5, 0.5) B B: (0, 1) Mahal(A,B) = 5 Mahal(A,C) = 4 B A

Common Properties of a Distance Distances, such as the Euclidean distance, have some well known properties. d(p, q)  0 for all p and q and d(p, q) = 0 only if p = q. (Positive definiteness) d(p, q) = d(q, p) for all p and q. (Symmetry) d(p, r)  d(p, q) + d(q, r) for all points p, q, and r. (Triangle Inequality) where d(p, q) is the distance (dissimilarity) between points (data objects), p and q. A distance that satisfies these properties is a metric

Common Properties of a Similarity Similarities, also have some well known properties. s(p, q) = 1 (or maximum similarity) only if p = q. s(p, q) = s(q, p) for all p and q. (Symmetry) where s(p, q) is the similarity between points (data objects), p and q.

Similarity Between Binary Vectors Common situation is that objects, p and q, have only binary attributes Compute similarities using the following quantities M01 = the number of attributes where p was 0 and q was 1 M10 = the number of attributes where p was 1 and q was 0 M00 = the number of attributes where p was 0 and q was 0 M11 = the number of attributes where p was 1 and q was 1 Simple Matching and Jaccard Coefficients SMC = number of matches / number of attributes = (M11 + M00) / (M01 + M10 + M11 + M00) J = number of 11 matches / number of not-both-zero attributes values = (M11) / (M01 + M10 + M11)

SMC versus Jaccard: Example q = 0 0 0 0 0 0 1 0 0 1 M01 = 2 (the number of attributes where p was 0 and q was 1) M10 = 1 (the number of attributes where p was 1 and q was 0) M00 = 7 (the number of attributes where p was 0 and q was 0) M11 = 0 (the number of attributes where p was 1 and q was 1) SMC = (M11 + M00)/(M01 + M10 + M11 + M00) = (0+7) / (2+1+0+7) = 0.7 J = (M11) / (M01 + M10 + M11) = 0 / (2 + 1 + 0) = 0

Cosine Similarity If d1 and d2 are two document vectors, then cos( d1, d2 ) = (d1  d2) / ||d1|| ||d2|| , where  indicates vector dot product and || d || is the length of vector d. Example: d1 = 3 2 0 5 0 0 0 2 0 0 d2 = 1 0 0 0 0 0 0 1 0 2 d1  d2= 3*1 + 2*0 + 0*0 + 5*0 + 0*0 + 0*0 + 0*0 + 2*1 + 0*0 + 0*2 = 5 ||d1|| = (3*3+2*2+0*0+5*5+0*0+0*0+0*0+2*2+0*0+0*0)0.5 = (42) 0.5 = 6.481 ||d2|| = (1*1+0*0+0*0+0*0+0*0+0*0+0*0+1*1+0*0+2*2) 0.5 = (6) 0.5 = 2.245 cos( d1, d2 ) = .3150

Extended Jaccard Coefficient (Tanimoto) Variation of Jaccard for continuous or count attributes Reduces to Jaccard for binary attributes

Correlation Correlation measures the linear relationship between objects To compute correlation, we standardize data objects, p and q, and then take their dot product

Visually Evaluating Correlation Scatter plots showing the similarity from –1 to 1.

Density Density-based clustering require a notion of density Examples: Euclidean density Euclidean density = number of points per unit volume Probability density Graph-based density

Euclidean Density – Cell-based Simplest approach is to divide region into a number of rectangular cells of equal volume and define density as # of points the cell contains

Euclidean Density – Center-based Euclidean density is the number of points within a specified radius of the point